Check the given equality: $\;$ $\sin^{-1} \left(\dfrac{4}{5}\right) + \sin^{-1} \left(\dfrac{5}{13}\right) + \sin^{-1} \left(\dfrac{16}{65}\right) = \dfrac{\pi}{2}$
$\begin{aligned} LHS & = \sin^{-1} \left(\dfrac{4}{5}\right) + \sin^{-1} \left(\dfrac{5}{13}\right) + \sin^{-1} \left(\dfrac{16}{65}\right) \\\\ & = \sin^{-1} \left(\dfrac{4}{5} \times \sqrt{1 - \dfrac{25}{169}} + \dfrac{5}{13} \times \sqrt{1 - \dfrac{16}{25}}\right) + \sin^{-1} \left(\dfrac{16}{65}\right) \\ & \left\{\because \;\; \sin^{-1} x + \sin^{-1} y = \sin^{-1} \left(x \sqrt{1 - y^2} + y \sqrt{1 - x^2}\right), \; 0 < x, y < 1\right\} \\\\ & = \sin^{-1} \left(\dfrac{4}{5} \times \dfrac{12}{13} + \dfrac{5}{13} \times \dfrac{3}{5}\right) + \sin^{-1} \left(\dfrac{16}{65}\right) \\\\ & = \sin^{-1} \left(\dfrac{63}{65}\right) + \sin^{-1} \left(\dfrac{16}{65}\right) \\\\ & = \sin^{-1} \left(\dfrac{63}{65} \times \sqrt{1 - \dfrac{256}{4225}} + \dfrac{16}{65} \times \sqrt{1 - \dfrac{3969}{4225}}\right) \\\\ & = \sin^{-1} \left(\dfrac{63}{65} \times \sqrt{\dfrac{3969}{4225}} + \dfrac{16}{65} \times \sqrt{\dfrac{256}{4225}}\right) \\\\ & = \sin^{-1} \left(\dfrac{63}{65} \times \dfrac{63}{65} + \dfrac{16}{65} \times \dfrac{16}{65}\right) \\\\ & = \sin^{-1} \left(\dfrac{4225}{4225}\right) \\\\ & = \sin^{-1} \left(1\right) \\\\ & = \dfrac{\pi}{2} = RHS \end{aligned}$